A shorter, simpler, stronger proof of the Meshalkin-Hochberg-Hirsch bounds on componentwise antichains

Abstract

Meshalkin's theorem states that a class of ordered p-partitions of an n-set has at most $\max \left(\genfrac{}{}{0ex}{}{n}{a_1,...,a_p}\right)$ members if for each k the k'th parts form an antichain. We give a new proof of this and the corresponding LYM inequality due to Hochberg and Hirsch, which is simpler and more general than previous proofs. It extends to a common generalization of Meshalkin's theorem and Erdos's theorem about r-chain-free set families.

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References in corpus (1)

• A Meshalkin theorem for projective geometries